Systems and methods for reducing an effect of a disturbance

ABSTRACT

A method for reducing an effect of a disturbance signal on an output of a dynamic system. The method includes generating an increment of the disturbance signal, and modifying an incremental signal input to the dynamic system based on the increment of the disturbance signal, thereby reducing the effect of the disturbance signal. According to one embodiment the method includes generating an increment by calculating a difference between two values sampled during consecutive sampling periods, wherein a first one of the two values is sampled during a first one of the consecutive sampling periods, and wherein a second one of the two values is sampled during a second one of the consecutive sampling periods, and wherein the two values are disturbances.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH & DEVELOPMENT

The invention described herein was made with Government support underContract No. N00019-96-C-0176 awarded by the Department of Defense. TheGovernment has certain rights to the invention.

BACKGROUND OF THE INVENTION

This invention relates generally to a dynamic system and moreparticularly to systems and methods for reducing an effect of adisturbance.

A dynamic system, such as a gas turbine, wind turbine, an engine, amotor, or a vehicle, has at least one input and provides at least oneoutput based on the at least one input. However, the dynamic system issubjected to a plurality of disturbances, which are inputs to thedynamic system that have an undesirable effect on the at least oneoutput.

BRIEF DESCRIPTION OF THE INVENTION

In one aspect, a method for reducing an effect of a disturbance signalon an output of a dynamic system is described. The method includesgenerating an increment of the disturbance signal, and modifying anincremental signal input to the dynamic system based on the increment ofthe disturbance signal.

In another aspect, a processor for reducing an effect of a disturbancesignal on an output of a dynamic system is described. The processor isconfigured to generate an increment of the disturbance signal, andmodify an incremental signal input to the dynamic system based on theincrement of the disturbance signal.

In yet another aspect, a method for attenuating is provided. The methodincludes attenuating, across a range of frequencies, an impact of adisturbance on a dynamic system.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an exemplary system for reducing an effectof a disturbance.

FIG. 2 is a block diagram of an exemplary dynamic apparatus which may beused with the system shown in FIG. 1.

FIG. 3 is a block diagram of an alternative embodiment of a dynamicsystem which may be used with the system shown in FIG. 1.

FIG. 4 shows an embodiment of a plurality of plots that may be used forreducing an effect of a disturbance.

FIG. 5 shows a plurality of exemplary graphs that may be used inreducing an effect of a disturbance.

FIG. 6 is a block diagram of an exemplary dynamic disturbance rejectionsystem (DDRS) that may be used with the system shown in FIG. 1.

FIG. 7 is a block diagram of an alternative embodiment of a DDRS, whichmay be used with the system shown in FIG. 1.

FIG. 8 illustrates a plurality of graphs showing system responseillustrating an exemplary effect of basic control without dynamicdisturbances.

FIG. 9 illustrates a plurality of graphs showing system responseillustrating an exemplary effect of disturbance signals without applyingthe method of reducing an effect of a disturbance.

FIG. 10 illustrates a plurality of graphs showing system responseillustrating an exemplary effect of disturbance signals and applying themethod of reducing an effect of a disturbance.

DETAILED DESCRIPTION OF THE INVENTION

FIG. 1 is a block diagram of an exemplary system 10 that may be used tofacilitate reducing an effect of a disturbance. In the exemplaryembodiment, system 10 includes a controller 12, a dynamic disturbancereduction system (DDRS) 14, and a dynamic system 16. As used herein, theterm controller is not limited to just those integrated circuitsreferred to in the art as a controller, but broadly refers to acomputer, a processor, a microcontroller, a microcomputer, aprogrammable logic controller, an application specific integratedcircuit, and other programmable circuits. Moreover, examples of DDRS 14include, but are not limited to, a controller, a computer, a processor,a microcontroller, a microcomputer, a programmable logic controller, anapplication specific integrated circuit, and other programmablecircuits. An example of dynamic system 16 includes, but is not limitedto, a water tank, a gas tank, an engine system, such as, an internalcombustion engine, a diesel engine, or a gas turbine used in powerplants or alternatively aircraft propulsion, and a vehicle. Examples ofthe vehicle include a car, an airplane, a truck, and a motorcycle.Dynamic system 16 can be a single-input-single-output (SISO) system oralternatively system 16 can be a multiple-input-multiple-output (MIMO)system. Dynamic system 16 may include a magnet, an electrical device, amechanical device, and/or a chemical substance. Dynamic system 16 may beimplemented in, but is not limited to being implemented in, aerospaceindustry, marine industry, paper industry, automotive industry, plasticindustry, food industry, and/or pharmaceutical industry.

Controller 12 receives from a supervisory system, such as a person orsupervisory computer, via an input device, a discrete controller inputsignal 18 or R_(k), which is a signal in a discrete time domain, andprocesses the input signal 18 or R_(k) to output a discrete controlleroutput signal 20 or v_(k), which is a signal in the discrete timedomain. The variable k is an integer. Each value of k represents asampling period t_(s) described below. For example, k=1 represents afirst sampling period and k=2 represents a second sampling period.Controller 12 receives controller input signal 18 from the person via aninput device, or from a supervisory computer across a communicationdevice, or from a supervisory algorithm in-situ with the discretecontroller 12. An example of the input device includes a mouse,keyboard, or any other analog or digital communication device. Anexample of a process performed by controller 12 on discrete controllerinput signal 18 includes integration, filtering, and/or determining arate of change of information within discrete controller input signal18. An example of discrete controller input signal 18 includes a signalrepresentative of a thrust demand, which is an amount of thrust, of apropulsion system and a power demand, which is an amount of power, of apower plant. Other examples of discrete controller input signal 18include a signal representative of a rate of change of an altitude and arate of change of speed. Examples of discrete controller output signal20 include a signal representative of a rate of the thrust demand, arate of change of the power demand, a rate of change of fuel flow, and arate of change of an exhaust nozzle area.

DDRS 14 receives discrete controller output signal 20 and a discretedisturbance signal 22 or d_(k), which is a signal in the discrete timedomain. DDRS 14 reduces an effect of discrete disturbance signal 22 on adynamic system output signal 24 or y_(k), which is a signal in thediscrete time domain, by generating a discrete DDRS output signal 26 oru_(k), which is a signal in the discrete time domain. Examples ofdiscrete dynamic system output signal 24 include a signal representativeof an engine pressure ratio (EPR) across an engine within dynamic system16, a thrust output by dynamic system 16, a speed of dynamic system 16,a power of dynamic system 16, and/or an increase or decrease in a levelof fluid within a fluid tank. Examples of discrete disturbance signal 22include a signal representative of a flow of air, a flow of fuel, a flowof water, or a flow of chemical into dynamic system 16, at least oneenvironmental ambient condition, such as humidity or condensation, dueto weather or an operating condition surrounding dynamic system 16, atemperature or alternatively pressure of the atmosphere surroundingdynamic system 16, a flow of energy from an actuator or alternatively aneffector into dynamic system 16, and/or a variable geometry, such as aplurality of variable stator vanes, a plurality of variable guide vanes,a plurality of variable by-pass ratios, which change basic physicalrelationships in dynamic system 16.

A sensor, such as a position sensor, a flow sensor, a temperature sensoror a pressure sensor, measures a parameter, such as a position, a flow,a temperature or alternatively a pressure, of a sub-system, such as atire or an engine, within dynamic system 16 to generate discretedisturbance signal 22. Alternatively, discrete disturbance signal 22 canbe estimated or calculated by an estimation algorithm executed by acontroller. For example, discrete disturbance signal 22 can be atemperature calculated or estimated, by the estimation algorithm and theestimation algorithm calculates or estimates the temperature by usinginformation from one or a combination of sensors including a speedsensor, a pressure sensor that senses a pressure at a location within oralternatively outside dynamic system 16, and a plurality of temperaturessensors that sense temperatures at a plurality of locations in dynamicsystem 16. It is noted that in an alternative embodiment, at least oneof controller 12 and DDRS 14 are coupled to a memory device, such as arandom access memory (RAM) or a read-only memory (ROM), and an outputdevice, such as a display, which can be a liquid crystal display (LCD)or a cathode ray tube (CRT).

FIG. 2 is a block diagram of a dynamic apparatus 50, which is an exampleof dynamic system 16. In the exemplary embodiment, exemplary dynamicapparatus 50 is a Multiple-Input Multiple-Output (MIMO) dynamicapparatus that receives a plurality of discrete dynamic apparatus inputsignals 52 and 54, receives a plurality of discrete dynamic apparatusdisturbance signals 56 and 58, and generates a plurality of discretedynamic apparatus output signals 60 and 62 based on input signals 52 and54 and/or disturbance signals 56 and 58. Each discrete dynamic apparatusinput signal 52 and 54 is an example of DDRS output signal 26 (shown inFIG. 1). Moreover, each discrete dynamic apparatus disturbance signal 56and 58 is an example of disturbance signal 22 (shown in FIG. 1), andeach discrete dynamic apparatus output signal 60 and 62 is an example ofdiscrete dynamic system output signal 24. It is noted that in analternative embodiment, dynamic apparatus 50 receives any number ofdiscrete dynamic apparatus input signals, and outputs any number ofdiscrete dynamic apparatus output signals based on the discrete dynamicapparatus input signals and discrete dynamic apparatus disturbancesignals.

FIG. 3 is a block diagram of an alternative embodiment of a dynamicsystem 16 that may be used with system 10 (shown in FIG. 1). Dynamicsystem 16 includes an integrator 100 and a plant 102. An example ofplant 102 includes an engine, such as a turbine engine or a car engine,and/or an electronic commutated motor. Integrator 100 receives discreteDDRS output signal 26, and integrates discrete DDRS output signal 26 togenerate a discrete integrator output signal 104. Plant 102 receivesdiscrete integrator output signal 104 and generates discrete dynamicsystem output signal 24 based on output signal 104 and disturbancesignal 22. For example, a turbine engine outputs thrust based on asignal representing an amount of fuel flow to the turbine engine andbased on its environmental operating conditions. As another example, avehicle engine outputs rotations per minute (RPM) of a vehicle based ona signal representing an amount of fuel flowing to the vehicle engineand the conditions in which the vehicle is operating.

DDRS 14 describes or models dynamic system 16 as a set of continuoustime nonlinear equations that may be represented as{dot over (x)} _(t) =ƒ(x _(t) ,u _(t) ,d _(t))  (1)y _(t) =h(x _(t) ,u _(t) ,d _(t))  (2)

where x_(t) is a state of a portion, such as a level of fluid within thefluid tank, an engine speed, or an engine temperature, of dynamic system16, t is continuous time, {dot over (x)}_(t) is a derivative, withrespect to the time t, of the state x_(t), u_(t) is a DDRS outputsignal, which is a continuous form of the discrete DDRS output signalu_(k), d_(t) is a disturbance signal, which is a continuous form of thediscrete disturbance signal d_(k), and y_(t) is a dynamic system outputsignal 24, which is a continuous form of the discrete dynamic systemoutput signal y_(k). For example, u_(k) is generated by sampling u_(t),d_(k) is generated by sampling d_(t), and y_(k) is generated by samplingy_(t). In one embodiment, f and h are each a nonlinear function. Anexample of the state x_(t) is a temperature of the turbine engine and/ora temperature of the car engine. Other examples of the state x_(t)include a speed of a rotating mass, a pressure, an amount of heat, anamount of potential energy, and/or an amount of kinetic energy containedin an energy storing element or device located within dynamic system 16.

DDRS 14 defines a nominal state value x _(t), which is a particularvalue of the state x_(t) at a reference time and defines an incrementalstate variable {tilde over (x)}_(t) for the state x_(t) as:x _(t) = x _(t) +{tilde over (x)} _(t).  (3)

where {tilde over (x)}_(t) is an increment to the nominal state value x_(t). Similarly, DDRS 14 defines a nominal input value ū_(t), which is aparticular value at the reference time of the DDRS output signal u_(t)and defines an incremental input variable ũ_(t) for the DDRS outputsignal u_(t) asu _(t) =ū _(t) +ũ _(t).  (4)

where ũ_(t) is an increment to the nominal input value ū_(t). Moreover,DDRS 14 defines a nominal output value, which is a particular value atthe reference time of the dynamic system output signal y_(t), anddefines an incremental output variable {tilde over (y)}_(t) for dynamicsystem output signal y_(t) as:y _(t) = y _(t) +{tilde over (y)} _(t).  (5)

where {tilde over (y)}_(t) is an increment to the nominal output value y_(t). Additionally, DDRS 14 defines a nominal disturbance value d _(t),which is a particular value at the reference time of the disturbancesignal d_(t), and defines an incremental disturbance variable {tildeover (d)}_(t) for disturbance signal d_(t) as:d _(t) = d _(t) +{tilde over (d)} _(t).  (6)

where {tilde over (d)}_(t) is an increment to the nominal disturbancevalue d _(t).

DDRS 14 linearizes the function f represented by equation (1) byapplying:

$\begin{matrix}\begin{matrix}{{\overset{.}{x}}_{t} = {{\overset{\overset{.}{\_}}{x}}_{t} + {\overset{\overset{.}{\sim}}{x}}_{t}}} \\{{{{{{{{{= {{f\left( {{\overset{\_}{x}}_{t},{\overset{\_}{u}}_{t},{\overset{\_}{d}}_{t}} \right)} + \frac{\partial f}{\partial x_{t}}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{x}}_{t}} + \frac{\partial f}{\partial u_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{u}}_{t}} + \frac{\partial f}{\partial d_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{d}}_{t}}\end{matrix} & (7)\end{matrix}$

where {dot over (x)}_(t) is a derivative, with respect to time t, of thestate x_(t),

is a derivative, with respect to time t, of the nominal state value x_(t),

is a derivative, with respect to time t, of the incremental statevariable

${{{\overset{\sim}{x}}_{t},\frac{\partial f}{\partial x_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function f, with respect to x_(t) and isevaluated at x _(t), d _(t), and ū_(t),

${\frac{\partial f}{\partial u_{t}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function ƒ, with respect to u_(t) and isevaluated at x _(t), d _(t), and ū_(t), and

${\frac{\partial f}{\partial d_{t}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function ƒ, with respect to d_(t) and isevaluated at x _(t), d _(t), and ū_(t).

Moreover, DDRS 14 expands the function h represented by equation (2) byapplying:

$\begin{matrix}\begin{matrix}{y_{t} = {{\overset{\_}{y}}_{t} + {\overset{\_}{y}}_{t}}} \\{{{{{{{{{= {{h\left( {{\overset{\_}{x}}_{t},{\overset{\_}{u}}_{t},{\overset{\_}{d}}_{t}} \right)} + \frac{\partial h}{\partial x_{t}}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{x}}_{t}} + \frac{\partial h}{\partial u_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{u}}_{t}} + \frac{\partial h}{\partial d_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{d}}_{t}}\end{matrix} & (8)\end{matrix}$

where

${\frac{\partial h}{\partial x_{t}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function h, with respect to x_(t) and isevaluated at x _(t), d _(t), and ū_(t),

${\frac{\partial h}{\partial u_{t}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function h, with respect to u_(t) and isevaluated at x _(t), d _(t), and ū_(t), and

${\frac{\partial h}{\partial d_{t}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$is a partial derivative of the function h, with respect to d_(t) and isevaluated at x _(t), d _(t), and ū_(t).

DDRS 14 represents a change in the state x_(t) as a function of a changein DDRS output signal u_(t) and a change in the disturbance signal d_(t)by representing the derivative

of the incremental state variable {tilde over (x)}_(t) as a function ofthe incremental disturbance variable {tilde over (d)}_(t) and a functionof the incremental input variable ũ_(t) as:

$\begin{matrix}{{{{{{{{{{{\overset{\overset{.}{\sim}}{x}}_{t} = {{f\left( {{\overset{\_}{x}}_{t},{\overset{\_}{u}}_{t},{\overset{\_}{d}}_{t}} \right)} + \frac{\partial f}{\partial x_{t}}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{x}}_{t}} + \frac{\partial f}{\partial u_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{u}}_{t}} + \frac{\partial f}{\partial d_{t}}}}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{\overset{\sim}{d}}_{t}} - {\overset{.}{\overset{\_}{x}}}_{t}} & (9)\end{matrix}$

DDRS 14 derives equation (9) by making {tilde over ({dot over (x)}_(t)the subject of equation (7). Moreover, DDRS 14 represents a change inthe dynamic system output signal y_(t) as a function of a change in theDDRS output signal u_(t) and a change in the disturbance signal d_(t) byrepresenting the incremental output variable {tilde over (y)}_(t) as afunction of the incremental disturbance variable {tilde over (d)}_(t)and a function of the incremental input variable ũ_(t) as:

$\begin{matrix}{{\overset{\sim}{y}}_{t} = \left. {{h\left( {{\overset{\_}{x}}_{t},{\overset{\_}{u}}_{t},{\overset{\_}{d}}_{t}} \right)} + \frac{\partial h}{\partial x_{t}}} \middle| {}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{{\overset{\sim}{x}}_{t} + \frac{\partial h}{\partial u_{t}}} \middle| {}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{{\overset{\sim}{u}}_{t} + \frac{\partial h}{\partial d_{t}}} \middle| {}_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}{{\overset{\sim}{d}}_{t} - {\overset{\_}{y}}_{t}} \right.} & (10)\end{matrix}$

DDRS 14 derives equation (10) by making {tilde over (y)}_(t) the subjectof equation (8).

DDRS 14 substitutes

${{\overset{.}{\overset{\_}{x}}}_{t} = {{f\left( {{\overset{\_}{x}}_{t},{\overset{\_}{u}}_{t},{\overset{\_}{d}}_{t}} \right)} = 0}},$substitutes A_(c) instead of

$\left. \frac{\partial f}{\partial x_{t}} \right|_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}},$substitutes B_(cu) instead of

$\left. \frac{\partial f}{\partial u_{t}} \right|_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}},$and B_(cd) instead of

$\left. \frac{\partial f}{\partial d_{t}} \right|_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$in equation (9) to generate:

$\begin{matrix}{{\overset{.}{\overset{\sim}{x}}}_{t} = {{A_{c}{\overset{\sim}{x}}_{t}} + {B_{cu}{\overset{\sim}{u}}_{t}} + {B_{cd}{\overset{\sim}{d}}_{t}} + {f.}}} & (11)\end{matrix}$

It is noted that

${\overset{.}{\overset{\_}{x}}}_{t} = 0$when x _(t) is a constant, an equilibrium solution, or a steady state ofthe portion of dynamic system 16. When x _(t) is a constant, regardlessof the time t, ƒ( x _(t),ū_(t), d _(t)) is also a constant regardless ofthe time t of evolution of representation of dynamic system 16, DDRS 14derives y _(t) from equation (2) by applying:y _(t) =h( x _(t) ,ū _(t) , d _(t))  (12)

Moreover, DDRS 14 substitutes y _(t) instead of h( x _(t),ū_(t), d _(t))a matrix C instead of

$\left. \frac{\partial h}{\partial x_{t}} \right|_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}},$a matrix D_(u) instead of

$\left. \frac{\partial h}{\partial u_{t}} \right|_{{\overset{\_}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}},$and a matrix D_(d) instead of

$\left. \frac{\partial h}{\partial d_{t}} \right|_{{\overset{.}{x}}_{t},{\overset{\_}{d}}_{t},{\overset{\_}{u}}_{t}}$of in equation (10) to generate:{tilde over (y)} _(t) =C{tilde over (x)} _(t) +D _(u) ũ _(t) +D _(d){tilde over (d)} _(t)  (13)

DDRS 14 generates a discrete time model of equation (11) by substituting{tilde over (x)}_(k) instead of {tilde over (x)}_(t), ũ_(k) instead ofũ_(t), {tilde over (d)}_(k) instead of {tilde over (d)}_(t) to generate:{tilde over (x)} _(k+1) =A{tilde over (x)} _(k) +B _(u) ũ _(k) +B _(d){tilde over (d)} _(k) +F _(k)  (14)

where DDRS 14 calculates a matrix A as being equal to I+A_(c)t_(s),calculates a matrix B_(u) to be equal to B_(cu)t_(s), a matrix B_(d) tobe equal to B_(cd)t_(s), F_(k) to be equal to f( x _(t),ū_(t), d_(t))t_(s), {tilde over (x)}_(k), an incremental discrete state, to beequal to a discrete form of {tilde over (x)}_(t), ũ_(k), an incrementaldiscrete DDRS output signal 26, to be equal to a discrete form of ũ_(t),and {tilde over (d)}_(k), an incremental discrete disturbance signal, tobe a discrete form of {tilde over (d)}_(t), t_(s) is a sampling time orthe sampling period, I is an identity matrix, and {tilde over (x)}_(k+1)is an incremental discrete state. Moreover {tilde over (x)}_(k+1) ofequation (14) can also be represented as a difference between a discretestate x_(k+1) and x_(k), where x_(k+1) is a discrete state of theportion of dynamic system 16 sampled during a sampling period k+1 and isgenerated one sampling period after x_(k) is generated, and {tilde over(x)}_(k) is the incremental discrete state. A microprocessor or acontroller samples x_(k) from x_(t) with the sampling period t_(s),samples y_(k) from y_(t) with the sampling period t_(s), samples u_(k)from u_(t) with the sampling period t_(s), and samples d_(k) from d_(t)with the sampling period t_(s). It is noted that d_(k), u_(k), x_(k),and y_(k) are samples that are sampled at the same time or during thesame sampling period k.

Furthermore, DDRS 14 generates a discrete time model of equation (13) bysubstituting {tilde over (x)}_(k) instead of {tilde over (x)}_(t), ũ_(k)instead of ũ_(t), and {tilde over (d)}_(k) instead of {tilde over(d)}_(t) in equation (13) to generate:{tilde over (y)} _(k) =C{tilde over (x)} _(k) +D _(u) ũ _(k) +D _(d){tilde over (d)} _(k)  (15)

where {tilde over (y)}_(k) is an incremental discrete dynamic systemoutput signal of dynamic system 16, where {tilde over (y)}_(k) isrepresented by a discrete form, y_(k)= y _(k)+{tilde over (y)}_(k), ofthe definition as provided in equation (5). If dynamic system 16 is arelative degree one system, DDRS 14 formulates a desired response ofdynamic system 16 as being a first order desired response. The relativedegree one system takes one sample period to change an output of dynamicsystem 16 based on an input to dynamic system 16. For example, when aninput to dynamic system 16 is u_(k), the relative degree one systemoutputs y_(k+1), which is a dynamic system output signal that is outputfrom dynamic system 16 one sample period after y_(k). is output fromdynamic system 16. DDRS 14 generates equations (16)-(25) based on therelative degree one system. A method similar to that of derivingequations (16)-(25) can be used to derive a plurality of equations for adynamic system of any relative degree, such as degrees two thru twenty.One form of the first order desired response is an integrator which canbe written as follows:{tilde over (y)} _(k+1) −{tilde over (y)} _(k) =t _(s) {tilde over (v)}_(k)  (16)

where {tilde over (y)}_(k+1) is a future incremental discrete dynamicsystem output from dynamic system 16 one sample after the current sample{tilde over (y)}_(k) is output from dynamic system 16, {tilde over(v)}_(k) is an incremental discrete controller output signal obtained asa difference between the discrete controller output signal v _(k) and anominal discrete controller output signal v_(k), which is a particularvalue of the discrete controller output signal v_(k) at the referencetime. The relative degree one dynamic system is an example of dynamicsystem 16.

DDRS 14 generates {tilde over (y)}_(k+1) from equation (15) as:{tilde over (y)} _(k+1) =C{tilde over (x)} _(k+1) +D _(u) ũ _(k+1) +D_(d) {tilde over (d)} _(k+1)  (17)

where {tilde over (d)}_(k+1), can also be represented as a differencebetween d_(k+1) and d_(k), where d_(k+1) is a discrete disturbancesignal input to dynamic system 16 during a sampling period k+1, and isgenerated one sampling period after d_(k) is generated, and ũ_(k+1) canalso be represented as a difference between u_(k+1) and u_(k), whereu_(k+1) is a discrete DDRS output signal output by DDRS 14 during asampling period k+1, and is generated one sampling period after u_(k) isgenerated. DDRS 14 substitutes {tilde over (x)}_(k+1) from equation (14)and D_(u)=0 for the relative degree one system into equation (17) togenerate:{tilde over (y)} _(k+1) =CA{tilde over (x)} _(k) +CB _(u) ũ _(k) +CB_(d) {tilde over (d)} _(k) +CF _(k) +D _(d) {tilde over (d)}_(k+1).  (18)

DDRS 14 further substitutes D_(u)=0 and equations (15) and (18) into thefirst desired response, represented by equation (16), to generate:CA{tilde over (x)} _(k) +CB _(u) ũ _(k) +CB _(d) {tilde over (d)} _(k)+D _(d) {tilde over (d)} _(k+1) +CF _(k) −C{tilde over (x)} _(k) −D _(d){tilde over (d)} _(k) =st{tilde over (v)} _(k)  (19)

DDRS 14 solves for ũ_(k) asũ _(k) =|CB _(u)|⁻¹ {t _(s) {tilde over (v)} _(k)+(C−CA){tilde over (x)}_(k)+(D _(d) −CB _(d)){tilde over (d)} _(k) −D _(d) {tilde over (d)}_(k+1) −CF _(k)}  (20)

DDRS 14 defines x_(k)= x _(k) within a relationship:x _(k) = x _(k) +{tilde over (x)} _(k)  (21)

to generate{tilde over (x)}_(k)=0  (22)

where x _(k) is a nominal discrete state value, which is a particularvalue at the reference time of the discrete state x_(k). Equation (21)is a discrete form of the relation expressed by equation (3).

DDRS 14 substitutes 2{tilde over (d)}_(k) instead of {tilde over(d)}_(k+1) in equation (20) and substitutes equation (22) into equation(20) to generate:ũ _(k) =|CB _(u)|⁻¹ {t _(s) {tilde over (v)} _(k) −CF _(k)+(−D _(d) −CB_(d)){tilde over (d)} _(k)}  (23)

DDRS 14 generates the discrete DDRS output signal u_(k) as being:u _(k) =u _(k−1) +ũ _(k)  (24)

where u_(k−1) is a discrete DDRS output signal output by DDRS 14 at k−1and generated one sampling period before u_(k) is output by DDRS 14, andũ_(k) is a discrete form in the discrete time domain of ũ_(t). DDRS 14substitutes equation (23) into equation (24), substitutes K₁ instead of|CB_(u)|⁻¹t_(s) in equation (24), K₃ instead of −|CB_(u)|⁻¹ C inequation (24), and K_(d) instead of |CB_(u)|⁻¹(−D_(d)−CB_(d)) inequation (24) to generate:u _(k) =u _(k−1) +K ₁ {tilde over (v)} _(k) +K ₃ F _(k) +K _(d) {tildeover (d)} _(k)  (25)

DDRS 14 computes K_(d) at least one of before and during energization ofdynamic system 16. For example, DDRS 14 computes K_(d) on-line in realtime while dynamic system 16 is being operated by a power source. Asanother example, DDRS 14 computes K_(d) off-line before dynamic system16 is provided power by the power source. DDRS 14 changes u_(k) at thesame time the disturbance signal d_(t) is input to dynamic system 16.Accordingly, an impact of the disturbance signal d_(t) on dynamic system16 is reduced.

As an alternative to formulating the first desired response, DDRS 14formulates one form of a second order desired response as:{tilde over (y)} _(k+2)−(1+α){tilde over (y)} _(k+1) +α{tilde over (y)}_(k)=(1−α)t _(s) {tilde over (v)} _(k)  (26)

where

${\alpha = \left( {1 - \frac{ts}{\tau}} \right)},$τ is a time constant of dynamic system 16, {tilde over (y)}_(k+2) is anincremental discrete dynamic system output signal, and {tilde over(y)}_(k+2) can also be represented as a difference between a dynamicsystem output signal y_(k+2) output by dynamic system 16 and y_(k+1),where y_(k+2) is sampled during a sampling period k+2 and is generatedone sampling period after y_(k+1) is generated. If dynamic system 16 isa relative degree two system, DDRS 14 formulates a desired response ofdynamic system 16 as being the second order desired response. Therelative degree two system takes two sample periods to change an outputof dynamic system 16 based on an input to dynamic system 16. Forexample, when an input to dynamic system 16 is u_(k), the relativedegree one system outputs y_(k+2), which is two sample periods aftery_(k). DDRS 14 generates equations (26)-(34) based on the relativedegree two system. The relative degree two system is an example ofdynamic system 16.

DDRS 14 substitutes CB_(u)=0 in equation (18) to output:{tilde over (y)} _(k+1) =CA{tilde over (x)} _(k) +CB _(d) {tilde over(d)} _(k) +CF _(k) +D _(d) {tilde over (d)} _(k+1)  (27)

DDRS 14 generates {tilde over (y)}_(k+2) from {tilde over (y)}_(k+1) ofequation (27) as being:{tilde over (y)} _(k+2) =CA{tilde over (x)} _(k+1) +CB _(d) {tilde over(d)} _(k+1) +CF _(k+1) +D _(d) {tilde over (d)} _(k+2)  (28)

where {tilde over (d)}_(k+2) is an incremental discrete disturbancesignal and can also be represented as a difference between a discretedisturbance signal d_(k+2) input to dynamic system 16 and d_(k+1), whered_(k+2) is sampled during a sampling period k+2 and is generated onesampling period after d_(k+1) is generated, and F_(k+1) is generatedduring a sampling period k+1, which is one sampling period after F_(k)is generated. DDRS 14 substitutes equation (14) into equation (28) togenerate:{tilde over (y)} _(k+2) =CA{A{tilde over (x)} _(k) +B _(u) ũ _(k) +B_(d) {tilde over (d)} _(k) +F _(k) }+CB _(d) {tilde over (d)} _(k+1) +CF_(k+1) +D _(d) {tilde over (d)} _(k+2)  (29)

DDRS 14 substitutes equations (15), (27), (29), and D_(u)=0 intoequation (26) to generate:

$\begin{matrix}{{{{CA}^{2}{\overset{\sim}{x}}_{k}} + {{CAB}_{u}{\overset{\sim}{u}}_{k}} + {{CAB}_{d}{\overset{\sim}{d}}_{k}} + {CAF}_{k} + {{CB}_{d}{\overset{\sim}{d}}_{k + 1}} + {CF}_{k + 1} + {D_{d}{\overset{\sim}{d}}_{k + 2}\mspace{11mu}\cdots} - {\left( {1 + \alpha} \right)\left\{ {{{CA}{\overset{\sim}{x}}_{k}} + {{CB}_{d}{\overset{\sim}{d}}_{k}} + {CF}_{k} + {D_{d}{\overset{\sim}{d}}_{k + 1}}} \right\}} + {\alpha\left\{ {{C{\overset{\sim}{x}}_{k}} + {D_{d}{\overset{\sim}{d}}_{k}}} \right\}}} = {\left( {1 - \alpha} \right)t_{s}{\overset{\sim}{v}}_{k}}} & (30)\end{matrix}$

DDRS 14 solves for ũ_(k) in equation (30) to output:

$\begin{matrix}{{\overset{\sim}{u}}_{k} = {{{CAB}_{u}}^{- 1}\left\{ {{\left( {1 - \alpha} \right)t_{s}{\overset{\sim}{v}}_{k}} + {\left\lbrack {{\left( {1 + \alpha} \right){CA}} - {CA}^{2} - {\alpha\; C}} \right\rbrack{\overset{\sim}{x}}_{k}} + {\left\lbrack {{\left( {1 + \alpha} \right)C} - {CA}} \right\rbrack F_{k}} - {{CF}_{k + 1}\mspace{11mu}\cdots} + {\left\lbrack {{\left( {1 + \alpha} \right){CB}_{d}} - {CAB}_{d} - {\alpha\; D_{d}}} \right\rbrack{\overset{\sim}{d}}_{k}} + {\left\lbrack {{\left( {1 + \alpha} \right)D_{d}} - {CB}_{d}} \right\rbrack{\overset{\sim}{d}}_{k + 1}} - {D_{d}{\overset{\sim}{d}}_{k + 2}}} \right\}}} & (31)\end{matrix}$

When x _(t), d _(t), ū_(t) and y _(t) are constant, with respect to thetime t, then:F _(k+1) =t _(s)ƒ( x _(k+1) ,ū _(k+1) , d _(k+1))=F _(k)  (32)

Equation (32) is calculated, by DDRS 14, based on values of thederivative {dot over (x)}_(t) or an estimation algorithm that computesthe derivative {dot over (x)}_(t) at the current sample x_(t). DDRS 14substitutes {tilde over (d)}_(k+1)=2{tilde over (d)}_(k), {tilde over(d)}_(k+2)=3{tilde over (d)}_(k), and equation (32) into equation (31)to generate:

$\begin{matrix}{{\overset{\sim}{u}}_{k} = {{{CAB}_{u}}^{- 1}\left\{ {{\left( {1 - \alpha} \right)t_{s}{\overset{\sim}{v}}_{k}} + {\left\lbrack {{\left( {1 + \alpha} \right){CA}} - {CA}^{2} - {\alpha\; C}} \right\rbrack{\overset{\sim}{x}}_{k}} + {\left\lbrack {{\alpha\; C} - {CA}} \right\rbrack F_{k}\mspace{11mu}\cdots} + {\left\lbrack {{\left( {\alpha - 1} \right){CB}_{d}} + {\left( {\alpha - 1} \right)D_{d}} - {CAB}_{d}} \right\rbrack{\overset{\sim}{d}}_{k}}} \right\}}} & (33)\end{matrix}$

DDRS 14 substitutes K₅ as being |CAB_(u)|⁻¹(1−α)t_(s), K₆ as being|CAB_(u)|⁻¹[(1+α)CA−CA²−αC], K₇ as being |CAB_(u)|⁻¹[αC−CA], K_(e) asbeing |CAB_(u)|⁻¹[(α−1)CB_(d)+(α−1)D_(d)−CAB_(d)], {tilde over(x)}_(k)=0, and equation (33) into equation (24) to output:u _(k) =u _(k−1) +K ₅ {tilde over (v)} _(k) +K ₇ F _(k) +K _(e) {tildeover (d)} _(k)  (34)

It is noted that {tilde over (x)}_(k)=0 when x _(k)=x_(k).

It is noted that in an alternative embodiment, if dynamic system 16 isof a relative degree n, DDRS 14 formulates an n^(th) order desiredresponse of dynamic system 16, where n is an integer greater than two.

DDRS 14 calculates the first, second, or alternatively the nth orderdesired response upon receiving a selection, via the input device,regarding a number, such as 1, 2 or alternatively n^(th), of a desiredresponse. As an example, upon receiving from the person via the inputdevice that a desired response has a first number, DDRS 14 applies anEuler's approximation to an integrator:{dot over (y)}_(t)=v_(t)  (35)

where v_(t) is a continuous form of v_(k).

to generatey _(k+1) −y _(k) =t _(s) v _(k)  (36)

DDRS 14 generates an incremental form of equation (36) to output thefirst order desired response. As another example, upon receiving fromthe person via the input device that a desired response is second order,DDRS 14 applies an Euler's approximation to a combination of anintegrator and the first order desired response of dynamic system 16.The combination is represented as:τÿ _(t) +{dot over (y)} _(t) =v _(t)  (37)

where {dot over (y)}_(t) is a derivative, with respect to the time t, ofy_(t), and {tilde over (y)}_(t) is a derivative, with respect to thetime t, of {dot over (y)}_(t). DDRS 14 applies an Euler's approximationto the combination to generate:y _(k+2)−(1+α)y _(k+1) +αy _(k)=(1−α)t _(s) v _(k)  (38)

DDRS 14 generates an incremental form of equation (38) to output thesecond desired response.

FIG. 4 shows an embodiment of a plurality of plots 300, 302, 304, and306 that may be used for reducing an effect of a disturbance. DDRS 14calculates and may generate plot 300, which is an example of d_(k)corresponding to the relative degree one system versus the time t.Moreover, DDRS 14 calculates and may generate plot 302, which is anexample of {tilde over (d)}_(k) plotted versus the time t and which isgenerated as a difference between d_(k) and d_(k−1), where d_(k−1) is adiscrete disturbance signal input to dynamic system 16 and measured by asensor, such as a temperature or a pressure sensor, one sampling periodbefore d_(k) is measured by the sensor. Moreover, DDRS 14 calculates andmay generate plot 304, which is an example of {tilde over (d)}_(k+1)plotted versus the time t and which is generated as a difference betweend_(k+1) and d_(k), where {tilde over (d)}_(k+1) is an incrementaldiscrete disturbance signal at k+1, where d_(k+1)1 is a discretedisturbance signal input to dynamic system 16 and measured by a sensor,such as a temperature or a pressure sensor, one sampling period befored_(k) is measured by the sensor. Additionally, DDRS 14 calculates andmay generate plot 306, which is an example of {tilde over (d)}_(k+2)plotted versus the time t and which is generated as a difference between{tilde over (d)}_(k+2) and {tilde over (d)}_(k+1), where {tilde over(d)}_(k+2) is an incremental discrete disturbance signal at k+2, whered_(k+2) is a discrete disturbance signal input to dynamic system 16 andmeasured by a sensor, such as a temperature or a pressure sensor, onesampling period before d_(k+1) is measured by the sensor.

FIG. 5 shows a plurality of exemplary graphs 310 and 312 that may beused to facilitate reducing an effect of a disturbance. Graph 310includes plots 302, 304, and 306, and graph 312 illustrates a plot of aratio 314, versus the time t, of plots 304 and 302, and a ratio 316,versus the time t, of plots 306 and 302. DDRS 14 generates ratios 314and 316. It is noted that for the relative degree one system, the ratio314 is two, and therefore, for the relative degree one system, {tildeover (d)}_(k+1)=2*{tilde over (d)}_(k). Moreover, it is noted that forthe relative degree one system, the ratio 316 is three, and therefore,for the relative degree one system, {tilde over (d)}_(k+2)=3*{tilde over(d)}_(k). In an alternative embodiment, for the relative degree nsystem, DDRS 14 determines {tilde over (d)}_(k+n) from {tilde over(d)}_(k) in a similar manner in which {tilde over (d)}_(k+1) and {tildeover (d)}_(k+2) are determined from {tilde over (d)}_(k).

FIG. 6 is a block diagram of an exemplary DDRS 500, which maybe usedwith system 10 (shown in FIG. 1) as a replacement for DDRS 14.Specifically, DDRS 500 may be used in system 10 to replace DDRS 14. DDRS500 includes a subtractor 502, a plurality of adders 504 and 506, aplurality of multipliers 508, 510, 512, 514, 516, and 520, and where K₁,K₃, and K_(d) are from equation (25).

Multiplier 508 receives the discrete disturbance signal d_(k) andmultiples d_(k) with 1/z, which is an inverse z-transform, to output thediscrete disturbance signal d⁻¹. Subtractor 502 receives the discretedisturbance signal d_(k) and the discrete disturbance signal d_(k−1),subtracts the discrete disturbance signal d_(k−1) from the discretedisturbance signal d_(k) to output the incremental discrete disturbancesignal {tilde over (d)}_(k). Multiplier 510 multiplies the incrementaldiscrete disturbance signal {tilde over (d)}_(k) with K_(d) to output amultiplier output signal 518. Multiplier 520 multiplies the derivative{dot over (x)}_(t) of the state x_(t) with t_(s) to output F_(k).Multiplier 514 receives F_(k) and multiplies F_(k) with K₃ to output amultiplier output signal 522. Multiplier 512 receives v_(k) andmultiplies v_(k) with K₁ to output a multiplier output signal 524. Adder504 receives multiplier output signals 518, 522, and 524, adds themultiplier output signals 518, 522, and 524 to generate an adder outputsignal 526, which is U_(k)−U_(k−1) in equation (25) and is equal toŨ_(k). Multiplier 516 receives u_(k) and multiplies u_(k) with 1/z tooutput u_(k−1). Adder 506 adds Ũ_(k) and u_(k−1) to output u_(k). Duringinitialization of DDRS 500, an initial value, such as zero, of u_(k), isprovided by the person to DDRS 14 via the input device. Upon receivingthe initial value and Ũ_(k), adder 506 outputs additional values ofu_(k). Dynamic system 16 receives u_(k) from DDRS 14 and u_(k) reducesan effect of d_(k) on y_(k).

FIG. 7 is a block diagram of an alternative embodiment of a DDRS 600,that may be used with system 10 (shown in FIG. 1) as a replacement forDDRS 14. DDRS 600 includes subtractor 502, adders 504, and 506, aplurality of multipliers 602, 604, and 606, and multipliers 508, 516,and 520.

Multiplier 602 multiplies the incremental discrete disturbance signal{tilde over (d)}_(k) with K_(e) to output a multiplier output signal608. Multiplier 606 receives F_(k) and multiplies F_(k) with K₇ tooutput a multiplier output signal 610. Multiplier 604 receives v_(k) andmultiplies v_(k) with K₅ to output a multiplier output signal 612. Adder504 receives multiplier output signals 608, 610, and 612, adds themultiplier output signals 608, 610, and 612 to generate an adder outputsignal 614, which is u_(k)−u_(k−1) in equation (34) and is equal toũ_(k). Dynamic system 16 receives u_(k) from DDRS 14 and u_(k) reducesan effect of d_(k) on y_(k). It is noted that K₁, K₃, K_(d), K₅, K₇, andK_(e) change based on a degree of dynamic system and based on otherfactors, such as the time constant τ.

FIG. 8 shows a plurality of exemplary graphs 700, 702, 704, 706, and 708including a plurality of exemplary outputs from dynamic system 16. Graph700 plots a disturbance signal 710 versus time t, graph 702 illustratesa plot of a disturbance signal 712 versus the time t, graph 704represents a dynamic system output signal 714 versus time t and adesired response 716 of dynamic system 16 versus time t. Moreover, graph706 illustrates a plot of a dynamic system output signal 718 versus timet and a desired response 720 of dynamic system 16 versus time t.Additionally, graph 708 illustrates a plot of a dynamic system outputsignal 722 versus time t and a desired response 724 of dynamic system 16versus time t. When disturbance signals 710 and 712 are input to dynamicsystem 16 and no disturbance is applied to dynamic system 16, dynamicsystem 16 generates dynamic system output signals 714, 718, and 722.Moreover, a difference between dynamic system output signal 714 anddesired response 716 is small and dynamic system output signal 714quickly converges to desired response 716. Additionally, a couplingbetween desired response 720 and dynamic system output signal 718 issmall, such as 9%-12%. Further, a coupling between desired response 724and dynamic system output signal 722 is small, such as 9%-12%.

FIG. 9 illustrates a plurality of exemplary graphs 750, 752, 754, 756,and 758. Graph 750 illustrates a plot of a disturbance signal 760 versusthe time t, graph 752 illustrates a plot of a disturbance signal 762versus the time t, and graph 754 illustrates a plot of a dynamic systemoutput signal 764 versus the time t and a desired response 766 of adynamic system, which is not coupled to DDRS 14, versus the time t.Moreover, graph 756 illustrates a plot of a dynamic system output signal770 versus the time t and a desired response 768 of a dynamic system,which is not coupled to DDRS 14, versus the time t. Additionally, graph758 illustrates a plot of a dynamic system output signal 774 versus thetime t and a desired response 772 of a dynamic system, which is notcoupled to DDRS 14, versus the time t. When disturbance signals 760 and762 are input to a dynamic system, which is not coupled to DDRS 14, thedynamic system generates dynamic system output signals 764, 770, and774. Moreover, a dynamic system output signal 764 slowly converges todesired response 766. Additionally, a coupling between desired response770 and dynamic system output signal 770 is large, such as 38%-42%.Further, a coupling between desired response 772 and dynamic systemoutput signal 774 is large, such as 38%-42%.

FIG. 10 illustrates a plurality of exemplary graphs 800, 802, 804, 806,and 808. Graph 800 illustrates a plot of a disturbance signal 810 versusthe time t, graph 802 illustrates a plot of a disturbance signal 812versus the time t, graph 804 illustrates a plot of a dynamic systemoutput signal 814 versus the time t and a desired response 816 ofdynamic system 16 versus the time t. Moreover, graph 806 illustrates aplot of a dynamic system output signal 818 versus the time t and adesired response 820 of dynamic system 16 versus the time t.Additionally, graph 808 illustrates a plot of a dynamic system outputsignal 822 versus the time t and a desired response 824 of dynamicsystem 16 versus the time t. When disturbance signals 810 and 812 areinput to dynamic system 16, dynamic system 16 generates dynamic systemoutput signals 814, 818, and 822. Moreover, a difference between dynamicsystem output signal 814 and desired response 816 is small and dynamicsystem output signal 814 quickly converges to desired response 816.Additionally, a coupling between desired response 820 and dynamic systemoutput signal 818 is small, such as 7%-8%. Further, a coupling betweendesired response 824 and dynamic system output signal 822 is small, suchas 7%-8%. It is evident from FIGS. 8 and 10 that the coupling betweendesired response 820 and dynamic system output signal 818 is similar tothat between desired response 720 and dynamic system output signal 718,and the coupling between desired response 824 and dynamic system outputsignal 822 is similar to that between desired response 724 and dynamicsystem output signal 722 when no disturbance is present.

Technical effects of the herein described systems and methods forreducing an effect of a disturbance include reducing an effect of thediscrete disturbance signal d_(k) on dynamic system output signal y_(k).The effect of the discrete disturbance signal d_(k) is reduced bygenerating an equation, such as equation (25) or (34), for theincremental discrete DDRS output signal Wk as a function of theincremental discrete disturbance signal {tilde over (d)}_(k), which is achange of a difference between the discrete disturbance signal d_(k) andthe discrete disturbance signal d_(k−1) By generating ü_(k) as afunction of {umlaut over (d)}_(k), changes, such as {umlaut over(d)}_(k), in the discrete disturbance signal d_(k) are considered inreducing the effect of the discrete disturbance signal d_(k) and DDRS 14attenuates an impact of the discrete disturbance signal d_(k) overdynamic system 16 over a broad frequency range, such as ranging from andincluding 0 hertz (Hz) to the closed loop bandwidth of the dynamicsystem. For systems such as gas turbine this range would be from 0 Hz to4 Hz, for electrical systems this range would be from 0 Hz to 10 kilohertz (KHz). Other technical effects of the systems and methods forreducing an effect of a disturbance include reducing coupling between adesired response of dynamic system 16 and dynamic system output signal24. Yet other technical effects include providing a quick convergence ofa dynamic system output signal 24 to a desired response. It is notedthat DDRS 14 does not wait to receive y_(k−1) to generate u_(k) andchanges u_(k) at the same time or during the same sampling period asd_(k) is received by dynamic system 16. Hence, an effect of d_(k) isreduced on dynamic system 16 before d_(k) enters and adversely affectsdynamic system 16.

While the invention has been described in terms of various specificembodiments, those skilled in the art will recognize that the inventioncan be practiced with modification within the spirit and scope of theclaims.

1. A method for reducing an effect of a disturbance signal on an outputof a dynamic system, said method comprising: generating an increment ofthe disturbance signal; and modifying an incremental signal input to thedynamic system based on the increment of the disturbance signal, therebyreducing the effect of the disturbance signal on the output of thedynamic system and; wherein generating an increment comprisescalculating a difference between two values sampled during consecutivesampling periods, wherein a first one of the two values is sampledduring a first one of the consecutive sampling periods, and wherein asecond one of the two values is sampled during a second one of theconsecutive sampling periods, and wherein the two values aredisturbances.
 2. A method in accordance with claim 1, wherein saidmodifying the incremental signal input to the dynamic system compriseschanging the incremental signal input to the dynamic system based on arelative degree of the dynamic system, wherein the relative degree isgreater than or equal to zero.
 3. A method in accordance with claim 1,wherein the disturbance signal comprises a signal that is configured notto affect the output of the dynamic system.
 4. A method in accordancewith claim 1 wherein the dynamic system is amultiple-input-multiple-output system.
 5. A method in accordance withclaim 1 further comprising estimating an additional signal generatedwithin a sampling period of the disturbance signal based on thedisturbance signal, wherein the additional signal represents adisturbance affecting the dynamic system.
 6. A method in accordance withclaim 1, wherein the first and second ones of the two values are inputto the dynamic system.
 7. A processor-based method that facilitatesreducing an effect of a disturbance signal on an output of a dynamicsystem, said processor-based method comprising: generating an incrementof the disturbance signal; and modifying an incremental signal input tothe dynamic system based on the increment of the disturbance signal,thereby attenuating the effect of the disturbance signal, and whereinsaid incremental signal input to the dynamic system is performed bycalculating u_k=u_k−1+K_(—)1˜v_K_(—)3F_k+K_d˜d_k, wherein u_k is asignal input to the dynamic system, u_K−1 is a signal input to thedynamic system one sampling period before u_k is input to the dynamicsystem, K_(—)1, K_(—)3, and K_d are matrices, ˜v_k is an incrementalsignal output by a controller to the processor, ˜d_k is an incrementaldisturbance input to the dynamic system, and F_k is a function of aplurality of values, at a reference time, of a disturbance input to thedynamic system, a state of a portion of the dynamic system, and a signalinput to the dynamic system.
 8. A processor in accordance with claim 7,wherein said processor is further configured to generate the incrementas a difference between two values sampled during consecutive samplingperiods, wherein a first one of the two values is sampled during a firstone of the consecutive sampling periods, and wherein a second one of thetwo values is sampled during a second one of the consecutive samplingperiods, and wherein the first and second ones of the two values aredisturbances.
 9. A processor in accordance with claim 7, wherein saidprocessor modifies the incremental signal input to the dynamic system bychanging the incremental signal input to the dynamic system based on arelative degree of the dynamic system, wherein the relative degree isgreater than or equal to zero.
 10. A processor in accordance with claim7, wherein the disturbance signal comprises a signal that is configurednot to affect the output of the dynamic system.
 11. A processor inaccordance with claim 7, wherein the dynamic system includes one of asingle-input-single-output system and a multiple-input-multiple-outputsystem.
 12. A processor in accordance with claim 7, wherein saidprocessor configured to estimate an additional signal generated within asampling period of the disturbance signal based on the disturbancesignal, wherein the additional signal represents a disturbance affectingthe dynamic system.
 13. A processor in accordance with claim 7, whereinthe incremental signal input to the dynamic system comprises adifference between two values sampled during consecutive samplingperiods, wherein a first one of the two values is sampled during a firstone of the consecutive sampling periods, wherein a second one of the twovalues is sampled during a second one of the consecutive samplingperiods, and wherein the first and second ones of the two values areinput to the dynamic system.
 14. A method for attenuating an impact of adisturbance on a dynamic system, comprising: generating an increment ofa disturbance signal; and modifying an incremental signal input to thedynamic system based on the increment of the disturbance signal, therebyattenuating the effect of the disturbance signal and adjusting an outputof said dyanmic system by calculatingu_k=u_k−1+K_(—)1˜v_k+K_(—)3F_k+K_d˜d_k, wherein u_k is a signal outputby a controller to the dynamic system for attenuating the disturbance,u_K−1 is a signal output by the controller to the dynamic system onesampling period before u_k is input to the dynamic system, K_(—)1,K_(—)3, and K_d are matrices, ˜v_k is an incremental signal output by aprocessor to the controller, ˜d_k is an incremental disturbance input tothe dynamic system, and F_k is a function of a plurality of values, at areference time, of a disturbance input to the dynamic system, a state ofa portion of the dynamic system, and a signal input to the dynamicsystem; wherein said attenuating is across a range of frequenciesranging from and including 0 Hz to the a bandwidth of the dynamicsystem.
 15. A method in accordance with claim 14, wherein saidattenuating the impact of the disturbance comprises deriving {dot over(x)}_(t)=ƒ(x_(t),u_(t),d_(t)), y_(t)=h(x_(t),u_(t),d_(t)),${{\overset{.}{\overset{\sim}{x}}}_{t} = {{A_{c}{\overset{\sim}{x}}_{t}} + {B_{cu}{\overset{\sim}{u}}_{t}} + {B_{c\; d}{\overset{\sim}{d}}_{t}} + f}},$and y _(t)=h( x _(t),ū_(t), d _(t)), wherein ƒ and h are nonlinearfunctions, x_(t) is a state of a portion of the dynamic system, t iscontinuous time, {dot over (x)}_(t) is a derivative, with respect to thetime t, of the state x_(t), u_(t) is a signal output by a controller tothe dynamic system, d_(t) is the disturbance input to the dynamicsystem, and each of A_(c), B_(cu), and B_(cd) are a matrix, {tilde over(x)}_(t) is an increment to a particular value x _(t) of the state at areference time, ũ_(t) is an increment to a particular value x _(t) ofu_(t) at the reference time, and {tilde over (d)}_(t) is an increment toa particular value of the disturbance d_(t) at the reference time.
 16. Amethod in accordance with claim 14, wherein said K_(d) is computed atleast one of before and during energization of the dynamic system.
 17. Amethod in accordance with claim 14, wherein said u_(k) is changed at thesame time the disturbance enters the dynamic system to attenuate theeffect of the disturbance on the dynamic system.